Integrable extensions of N = 2 supersymmetric KdV hierarchy associated with the nonuniqueness of the roots of the Lax operator

We present new supersymmetric integrable extensions of the a = 4, N = 2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generates a new hierarchy of integrable equations, for which we investigate the Hamiltonian structure. In a special case our system describes the interaction of the KdV equation with the two MKdV equations.     

The entropy function for the extremal Kerr-(anti-)de Sitter black holes

Based on Sen's entropy function formalism, we consider the Bekenstein-Hawking entropy of the extremal Kerr-(anti-)de Sitter black holes in 4-dimensions. Unlike the extremal Kerr black hole case with flat asymptotic geometry, where the Bekenstein-Hawking entropy S is proportional to the angular momentum J, we get a quartic algebraic relation between S and J by using the known solution to the Einstein equation. We recover the same relation in the entropy function formalism. Instead of full geometry, we write down an ansatz for the near horizon geometry only. The exact form of the unknown functions and parameters in the ansatz are obtained by solving the differential equations which extremize the entropy function. The results agree with the nontrivial relation between S and J.We also study the Gauss-Bonnet correction to the entropy exploiting the entropy function formalism. We show that the term, though being topological thus does not affect the solution, contributes a constant addition to the entropy because the term shifts the Hamiltonian by that amount.     

Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe

We study the semiclassical limit of the (generalized) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in H ^s to the solution of the Hopf equation, provided the initial data belongs to H ^s , ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities.     

Hints on integrability in the Wilsonian/holographic renormalization group

The Polchinski equations for the Wilsonian renormalization group in the D-dimensional matrix scalar field theory can be written at large N in a Hamiltonian form. The Hamiltonian defines evolution along one extra holographic dimension (energy scale) and can be found exactly for the subsector of Trϕ ⁿ (for all n) operators. We show that at low energies independently of the dimensionality D the Hamiltonian system in question reduces to the integrable effective theory. The obtained Hamiltonian system describes large wavelength KdV type (Burger-Hopf) equation with an external potential and is related to the effective theory obtained by Das and Jevicki for the matrix quantum mechanics.     

Closed Geodesics and Billiards on Quadrics Related to Elliptic KdV Solutions

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards in ⁿ . Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero system.     

The Bi-Hamiltonian Structure and New Solutions of KdV6 Equation

We show that the KdV6 equation and the nonholonomic perturbation of bi-Hamiltonian system of KdV hierarchy recently studied in Karasu-Kalkanli et al. (J Math Phys 49:073516, 2008) and Kupershmidt (Phys Lett A 372:2634–2639, 2008) are equivalent to the Rosochatius deformation of KdV equation and KdV hierarchy with self-consistent sources (RD-KdVESCS, RD-KdVHSCS), respectively, recently presented in Yao and Zeng (J Phys A Math Theor 41:295205, 2008). The t-type bi-Hamiltonian formalisms of KdV6 equation and RD-KdVHSCS are constructed by taking x as evolution parameter. Some new solutions of KdV6 equation, such as soliton, positon and negaton solution, are presented.     

Chern–Simons theory with vector fermion matter

We study three-dimensional conformal field theories described by U(N) Chern?CSimons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger?CDyson equation in light-cone gauge, we compute the exact planar free energy of the theory at finite temperature on ?² as a function of the ??t?Hooft coupling ??=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |??|=1; the conformal theory does not exist for |??|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three-point functions up to two loops. We also discuss a light-cone Hamiltonian formulation of this theory where a W _?? algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.     

Hirota-Satsuma dynamics as a non-relativistic limit of KdV equations

We consider a system of two coupled KdV equations (one for left-movers, the other for right-movers) and investigate its ultra-relativistic and non-relativistic limits in the sense of BMS₃/GCA₂ symmetry. We show that there is no local ultra-relativistic limit of the system with positive energy, regardless of the coupling constants in the original relativistic Hamiltonian. By contrast, local non-relativistic limits with positive energy exist, provided there is a non-zero coupling between left- and right-movers. In these limits, the wave equations reduce to Hirota-Satsuma dynamics (of type iv) and become integrable. This is thus a situation where input from high-energy physics contributes to nonlinear science — in this case, uncovering the limiting relation between integrable structures of KdV and Hirota-Satsuma.     

The behavior of the weyl function in the zero-dispersion KdV limit

The moment formulas that globally characterize the zero-dispersion limit of the Korteweg-deVries (KdV) equation are known to be expressed in terms of the solution of a maximization problem. Here we establish a direct relation between this maximizer and the zero-dispersion limit of the logarithm of the Jost functions associated with the inverse spectral transform. All the KdV conserved densities are encoded in the spatial derivative of these functions, known as Weyl functions. We show the Weyl functions are densities of measures that converge in the weak sense to a limiting measure. This limiting measure encodes all of the weak limits of the KdV conserved densities. Moreover, we establish the weak limit of spectral measures associated with the Dirichlet problem. Dedicated to Peter Lax on his 70^th birthday     

Photodissociation of NaH using time-dependent fourier grid method

We have solved the time dependent Schrödinger equation by using the Chebyshev polynomial scheme and Fourier grid Hamiltonian method to calculate the dissociation cross section of NaH molecule by 1-photon absorption from the X ¹Σ⁺ state to the B ¹Π state. We have found that the results differ significantly from an earlier calculation [1] although we have used the same set of potential energy curves [2].     

On an isomonodromy deformation equation without the Painlevé property

We show that the fourth-order nonlinear ODE which controls the pole dynamics in the general solution of equation P _I ² compatible with the KdV equation exhibits two remarkable properties: (1) it governs the isomonodromy deformations of a 2 × 2 matrix linear ODE with polynomial coefficients, and (2) it does not possess the Painlevé property. We also study the properties of the Riemann-Hilbert problem associated to this ODE and find its large-t asymptotic solution for physically interesting initial data.     

Universal scaling behaviour for iterated maps in the complex plane

According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ _n ^th iterates of the map, whereQ _n is then ^th Fibonacci number.     

Extended supersymmetries and the Dirac operator

We consider supersymmetric quantum mechanical systems in arbitrary dimensions on curved spaces with nontrivial gauge fields. The square of the Dirac operator serves as Hamiltonian. We derive a relation between the number of supercharges that exist and restrictions on the geometry of the underlying spaces as well as the admissible gauge field configurations. From the superalgebra with two or more real supercharges we infer the existence of integrability conditions and obtain a corresponding superpotential. This potential can be used to deform the supercharges and to determine zero modes of the Dirac operator. The general results are applied to the Kähler spaces CPⁿ.     

Connection between the KdV equation and the two-dimensional Ising model

We construct the one-parameter family of solutions to d²w/dzsu2 = zw + 2w³ that tend to zero for z → +∞ by specializing an equation previously solved in connection with the two-dimensional Ising model. These solutions are intimately related to the KdV equation.     

Ideal hydrodynamics outside and inside a black hole: Hamiltonian description in Painlevé-Gullstrand coordinates

We show that when the Painlevé-Gullstrand coordinates are used in their Cartesian version, the Hamiltonian of relativistic ideal hydrodynamics in the vicinity of a nonrotating black hole differs by only one simple term from the corresponding Hamiltonian in a flat spacetime. The interior region of the black hole is also described in a unified way, because there is no singularity on the event horizon in Painlevé-Gullstrand coordinates. We present the exact solution describing the steady accretion of extremely hard matter (? ∝ n ²) onto a moving black hole up to the central singularity. In the local induction approximation, we derive the equation of motion for a thin vortex filament against the background of such an accretion flow. We explicitly calculate the Hamiltonian for a fluid with an ultrarelativistic equation of state, ? ∝ n ^4/3, and solve the problem of a centrally symmetric steady flow of such matter.     

Evolution equations and sl(2, R)-valued Bäcklund forms

Evolution equations, in one variable, which are determined by a Bäcklund equation, on sl(2, R), are classified. A geometrical setting is given for the Gardner construction of conserved quantities. A relation between different constructions of conserved quantities for the KdV equation is found. It is shown that for all evolution equations obtained in this classification, these constructions reduce essentially to the KdV case. The second Poisson structure is derived and a Kac-Moody formulation is given, which allows one to deal with all higher order KdV-flows.     

Relativistic Point Dynamics and Einstein Formula as a Property of Localized Solutions of a Nonlinear Klein-Gordon Equation

Einstein’s relation E = Mc ² between the energy E and the mass M is the cornerstone of the relativity theory. This relation is often derived in a context of the relativistic theory for closed systems which do not accelerate. By contrast, the Newtonian approach to the mass is based on an accelerated motion. We study here a particular neoclassical field model of a particle governed by a nonlinear Klein-Gordon (KG) field equation. We prove that if a solution to the nonlinear KG equation and its energy density concentrate at a trajectory, then this trajectory and the energy must satisfy the relativistic version of Newton’s law with the mass satisfying Einstein’s relation. Therefore the internal energy of a localized wave affects its acceleration in an external field as the inertial mass does in Newtonian mechanics. We demonstrate that the “concentration” assumptions hold for a wide class of rectilinear accelerating motions.     

Non-linear diffusion in Hamiltonian systems exhibiting chaotic motion

The exact evolution equation for the angle averaged phase space density in action-angle space is derived from the Liouville equation using projection operator techniques. This equation involves a correlation function of the initial value of the phase space density with the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian with itself. Each of these correlation functions develops in time with angle projected dynamics. We show their relation to the correlation functions which develop in time with usual Hamiltonian dynamics. These correlation functions are then studied in the standard model of Chirikov, and we conclude that they behave as $\text{e}{}^{\mathrm{-\sigma t}}\text{cos}\text{(Ωt + φ)}$ in regions of irregular motion. We conjecture that angle averaged correlation functions behave this way in general, and we give an argument based on the mixing property of the Hamiltonian system. Our argument goes beyond the usual mixing, so we regard it as a quasi-mixing hypothesis. Under this hypothesis the equation for the angle averaged phase space density becomes a diffusion equation which incorporates much of the non-linear dynamics of Hamiltonian systems exhibiting chaotic motion.     

Harper Operator,Fermi Curves and Picard–Fuchs Equation

This paper is a continuation of the work on the spectral problem of the Harper operator using algebraic geometry. We continue to discuss the local monodromy of algebraic Fermi curves based on Picard–Lefschetz formula. The density of states over approximating components of Fermi curves satisfies a Picard–Fuchs equation. By the property of Landen transformation, the density of states has a Lambert series as the quarter period. A q-expansion of the energy is derived from a mirror map as in the B-model.     

Distribution of periodic orbits for the Casati-Prosen map on rational lattices

We study numerically the periodic orbits of the Casati-Prosen map, a two-parameter reversible map of the torus, with zero entropy. For rational parameter values, this map preserves rational lattices, and each lattice decomposes into periodic orbits. We consider the distribution function of the periods over prime lattices, and its dependence on the parameters of the map. Based on extensive numerical evidence, we conjecture that, asymptotically, almost all orbits are symmetric, and that for a set of rational parameters having full density, the distribution function approaches the gamma-distribution R(x)=1−e^−x(1+x). These properties, which have been proved to hold for random reversible maps, were previously thought to require a stronger form of deterministic randomness, such as that displayed by rational automorphisms over finite fields. Furthermore, we show that the gamma-distribution is the limit of a sequence of singular distributions which are observed on certain lines in parameter space. Our experiments reveal that the convergence rate to R is highly non-uniform in parameter space, being slowest in sharply-defined regions reminiscent of resonant zones in Hamiltonian perturbation theory.